Thermodynamic formalism for countable Markov shifts

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of φ =0 is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let L_φ denote the Ruelle operator for φ. We offer a definition of positive recurrence for φ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a σ-finite measure ν, a continuous function $h>0$ and λ >0 such that L_φ *ν =λ ν, L_φ h=λ h and λ ⁻ⁿL_φ ⁿf→ h∫ f\,dν for suitable functions f. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form φ =log f( 1x₀+ 1x₁+ ), where f is a suitable function on a suitable shift X.